The characteristic polynomial on compact groups with Haar measure : some equalities in law

TL;DR

This paper explores equalities in law for determinants of unitary matrices with Haar measure, showing they can be decomposed into products of independent variables, with implications for understanding spectral properties.

Contribution

It introduces a novel decomposition of the characteristic polynomial on compact groups with Haar measure, linking recursive measure decomposition and orthogonal polynomial techniques.

Findings

Decomposition of $Z_N$ into independent random variables.

Explicit laws for the factors in the decomposition.

Connections between determinants of submatrices and spectral properties.

Abstract

This note presents some equalities in law for $Z_N:=\det(\Id-G)$, where $G$ is an element of a subgroup of the set of unitary matrices of size $N$, endowed with its unique probability Haar measure. Indeed, under some general conditions, $Z_N$ can be decomposed as a product of independent random variables, whose laws are explicitly known. Our results can be obtained in two ways : either by a recursive decomposition of the Haar measure or by previous results by Killip and Nenciu on orthogonal polynomials with respect to some measure on the unit circle. This latter method leads naturally to a study of determinants of a class of principal submatrices.